全网最全Python图算法

图几个关键概念

⚫图可以表示为一个二元组𝑮 =< 𝑽, 𝑬 >，其中

𝑽表示非空顶点集，其元素称为顶点(Vertex)
𝑬表示边集，其元素称为边(Edge)
𝒆 = 𝒖, 𝒗 表示一条边，其中𝒖 ∈ 𝑽, 𝒗 ∈ 𝑽, 𝒆 ∈ 𝑬

⚫无向图

若顶点𝒖到𝒗之间的边没有方向，则称这条边为无向边（Edge），用无序偶对（𝒖,𝒗）来表示。
如果图中任意两个顶点之间的边都是无向边，则称该图为无向图。

⚫有向图

若顶点𝒖到𝒗之间的边有方向，则称这条边为有向边，也称为弧（Arc）。用有序偶<𝒖,𝒗>来表示，𝒖称为弧尾（Tail），𝒗称为弧头（Head）。
如果图中任意两个顶点之间的边都是有向边，则称该图为有向图（Directed graphs）。
⚫ 有向图顶点的度分为入度和出度
- 顶点𝒖的入度：起点为𝒖的边数
- 顶点𝒖的出度：终点为𝒖的边数

⚫ 相邻(Adjacent)

边𝒖, 𝒗 连接的顶点𝒖和𝒗相邻

⚫ 关联(Incident)

边𝒖, 𝒗 和其连接的顶点𝒖(或𝒗)相互关联

⚫ 顶点的度(Degree of a Vertex)

顶点𝒗的度𝒅𝒆𝒈(𝒗)是𝒗关联的边数

⚫ 图的度(Degree of a Graph)

图𝑮 =< 𝑽, 𝑬 >的度，是图各顶点的度之和，𝒅𝒆𝒈(𝑮) = σ𝒗∈𝑽 𝒅𝒆𝒈(𝒗)

⚫握手定理(Handshaking Lemma)

无向图的度是边数的两倍， 𝒅𝒆𝒈(𝑮) = 𝟐|𝑬|

⚫ 路径(Path)

图中一个的顶点序列< 𝒗𝟎, 𝒗𝟏, … , 𝒗𝒌 >称为𝒗𝟎到𝒗𝒌的路径
路径包含顶点𝒗𝟎, 𝒗𝟏, … , 𝒗𝒌和边𝒗𝟎, 𝒗𝟏 , 𝒗𝟏, 𝒗𝟐 , … , (𝒗𝒌−𝟏, 𝒗𝒌)
存在路径< 𝒗𝟎, 𝒗𝟏, … , 𝒗𝒌 > ，则𝒗𝟎可达𝒗𝒌
如果𝒗𝟎, 𝒗𝟏, … , 𝒗𝒌互不相同，则该路径是简单的

⚫环路(Cycle)

如果路径< 𝒗𝟎, 𝒗𝟏, … , 𝒗𝒌 >中𝒗𝟎 = 𝒗𝒌且至少包含一条边，则该路径构成环路
如果𝒗𝟏, 𝒗𝟐, … , 𝒗𝒌互不相同，则该环路是简单的

⚫连通(Connectivity)

如果图的任意对顶点互相可达，则称该图是连通的，反之称为非连通。

⚫ 通分量(Connected Components)

根据是否连通将顶点进行分组，相互可达的顶点集称为连通分量

⚫ 子图(Subgraph)

如果𝑽′ ⊆ 𝑽, 𝑬′ ⊆ 𝐄，则称图𝑮’ =< 𝑽’, 𝑬’ >是图𝑮的一个子图

⚫ 生成子图(Spanning Subgraph)

如果𝑽′ = 𝑽, 𝑬′ ⊆ 𝐄，则称图𝑮’ =< 𝑽’, 𝑬’ >是图𝑮的一个生成子图

⚫ 树(Tree)

连通、无环图𝑻 =< 𝑽𝑻, 𝑬𝑻 >，树有|𝑽𝑻| − 𝟏条边

⚫ 森林(Forest)

一至多棵树组成的无环图

图的存储形式(边、顶点的数据结构)

⚫ 图𝑮 =< 𝑽, 𝑬 > ，其邻接链表由|𝑽|条链表的数组构成

每个顶点有一条链表，包含所有与其相邻的顶点
𝑨𝒅𝒋[𝒂] ={𝒃, 𝒅}; 𝑨𝒅𝒋[𝒃] ={ 𝒂, 𝒄, 𝒅, 𝒇 }; 𝑨𝒅𝒋[𝒄] ={𝒃, 𝒇 }; …
空间大小𝑶(|𝑽| + |𝑬|)

为了实现稀疏连接的图，更高效的方式是使用邻接表。在邻接表实现中，我们为图对象的所有顶点保存一个主列表，同时为每一个顶点对象都维护一个列表，其中记录了与它相连的顶点。在对Vertex类的实现中，我们使用字典（而不是列表），字典的键是顶点，值是权重。

邻接表的优点是能够紧凑地表示稀疏图。此外，邻接表也有助于方便地找到与某一个顶点相连的其他所有顶点。

在Python中，通过字典可以轻松地实现邻接表。我们要创建两个类：Graph类存储包含所有顶点的主列表，Vertex类表示图中的每一个顶点。

Vertex使用字典connectedTo来记录与其相连的顶点，以及每一条边的权重。
以下Vertex类的实现，其构造方法简单地初始化id（它通常是一个字符串），以及字典connectedTo,以及其他算法所需要的辅助参数。

addNeighbor方法添加从一个顶点到另一个的连接。
getConnections方法返回邻接表中的所有顶点，由connectedTo来表示。
getWeight方法返回从当前顶点到以参数传入的顶点之间的边的权重。
setPred()\getPred() 设置\获取前驱顶点。
setColor()\getColor() 设置\获取顶点状态（颜色）。
setDistance()\getDistance() 设置\获取顶点的距离。

class Vertex:
    '''
    创建一个空顶点
    '''
    def __init__(self,key):
        self.id = key
        self.connectedTo = {}
        self.color = 'white'
        self.dist = inf
        self.pred = None
        self.disc = 0
        self.fin = 0
    def addNeighbor(self, nbr, weight = 0):
        '''
        添加邻接的顶点
        :param nbr: 邻接的顶点
        :param weight: 默认为0
        :return:
        '''
        self.connectedTo[nbr] = weight

    def getConnections(self):
        '''
        获取邻接的所以顶点
        :return:
        '''
        return self.connectedTo.keys()
    def getId(self):
        '''
        获取自身顶点的值
        :return:
        '''
        return self.id
    def getWeight(self, nbr):
        '''
        两顶点连接边的权重
        :param nbr:
        :return:
        '''
        return self.connectedTo[nbr]

    def setColor(self, color):
        '''
        设置顶点的颜色（一般设置为白色表示未访问，灰色表示待处理，黑色表示访问完成）。
        :param color: 字符串、或者int 例如 color = 'white' color = -1
        :return:
        '''
        self.color = color

    def getColor(self):
        '''
        获取顶点的颜色，（一般设置为白色表示未访问，灰色表示待处理，黑色表示访问完成）
        :return:
        '''
        return self.color

    def setDistance(self, d):
        '''
        设置顶点距离，（不同算法有不同的意义）如单源最短路径表示源点到该点最短距离，最小生成树表示离该顶点最小的权重。
        :param d:
        :return:
        '''
        self.dist = d
        
    def getDistance(self):
        '''
        获取该顶点的距离，（不同算法有不同的意义）如单源最短路径表示源点到该点最短距离，最小生成树表示离该顶点最小的权重。
        :return:
        '''
        return self.dist


    def setPred(self, p):
        '''
        设置该顶点的前驱顶点
        :param p:
        :return:
        '''
        self.pred = p

    def getPred(self):
        '''
        设置该顶点的前驱顶点
        :return:
        '''
        return self.pred

    ##以下方法用得不多，帮助理解。

    def setDiscovery(self, dtime):
        '''
        设置顶点发现时间。
        :param dtime:
        :return:
        '''
        self.disc = dtime
    def getDiscovery(self):
        '''
        获取顶点发现时间。
        :return:
        '''
        return self.disc

    def setFinish(self, ftime):
        '''
        设置顶点完成时间。
        :param ftime:
        :return:
        '''
        self.fin = ftime

    def getFinish(self):
        '''
        获取顶点完成时间。
        :return:
        '''
        return self.fin

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118

图的抽象数据类型

图的抽象数据类型由下列方法定义。

Graph()新建一个空图。
addVertex(vert)向图中添加一个顶点实例。
addEdge(fromVert, toVert)向图中添加一条有向边，用于连接顶点fromVert和toVert。
addEdge(fromVert, toVert, weight)向图中添加一条带权重weight的有向边，用于连接顶点fromVert和toVert。
getVertex(vertKey)在图中找到名为vertKey的顶点。
getVertices()以列表形式返回图中所有顶点。
in通过vertex in graph这样的语句，在顶点存在时返回True，否则返回False。
reverse()生成逆向图。
InDegreeUpdate 入度更新。
OutDegreeUpdate 出度更新。
bfs() 对图进行广度优先搜索。
dfs() 对图进行深度优先搜索。
prim() 生成最小生成树。
kruskal() 生成最小生成树。
dijkstra() 单源最短路径。
bellman_ford() 单源最短路径。
dfs_judge_cycle() 深度优先搜索进行环路判断。
topological_sort_bfs() 宽度优先搜索生成拓扑排序。
topological_sort_dfs() 深度优先搜索生成拓扑排序。
Strongly_Connected_Component()生成强联通分量。


class Graph:
    '''
    新建一个空图。
    '''
    def __init__(self):
        '''
        初始化邻接表，顶点个数
        '''
        self.vertList = {}
        self.numVertices = 0
        self.time = 0
    def addVertex(self, key):
        '''
        向图中添加一个顶点实例
        :param key:
        :return: None
        '''
        self.numVertices = self.numVertices + 1
        newVertex = Vertex(key)
        self.vertList[key] = newVertex
        return newVertex

    def getVertex(self, n):
        '''
        在图中找到名为n的顶点
        :param n:
        :return:
        '''
        if n in self.vertList:
            return self.vertList[n]
        else:
            return None
    def __contains__(self,n):
        '''
        对in进行重载
        :param n:
        :return:
        '''
        return n in self.vertList
    def addEdge(self, f, t, weight = 0):
        '''
        )向图中添加一条带权重cost的有向边，用于连接顶点f和t
        :param f:
        :param t:
        :param weight:
        :return: None
        '''
        if f not in self.vertList:
            nv = self.addVertex(f)
        if t not in self.vertList:
            nv = self.addVertex(t)
        self.vertList[f].addNeighbor(self.vertList[t],weight)

    def getVertics(self):
        '''
        以列表形式返回图中所有顶点
        :return: {}.keys()
        '''
        return self.vertList.keys()
    def __iter__(self):
        '''
        对迭代进行重载，方便遍历顶点邻接表
        :return:
        '''
        return iter(self.vertList.values())

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67

图的遍历

广度优先搜索

广度优先搜索伪代码
在这里插入图片描述

在这里插入图片描述

    def bfs(self,start = None):
        '''
        广度优先搜索
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]
        start.setDistance(0)
        start.setPred(None)
        vertQueue = queue.Queue()
        vertQueue.put(start)
        out = []
        while vertQueue.empty() == False:
            currentVert = vertQueue.get()
            for nbr in currentVert.getConnections():
                if nbr.getColor() == 'white':
                    nbr.setColor('gray')
                    nbr.setDistance(currentVert.getDistance() + 1)
                    nbr.setPred(currentVert.getId())
                    vertQueue.put(nbr)
            currentVert.setColor('black')
            out.append(currentVert.id)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

深度优先搜索

深度优先搜索伪代码
DFS(𝑮)
在这里插入图片描述

DFS-Visit(𝑮, 𝒗)
在这里插入图片描述

DFS(𝑮)

  def dfs(self):
        '''
        深度优先搜索
        :return:
        '''
        out = []
        self.time = 0
        for aVertex in self:
            aVertex.setColor('white')
            aVertex.setPred(None)
        for aVertex in self:
            if aVertex.getColor() == 'white':
                self.dfsvisit(aVertex,out)
        return out

    def dfsvisit(self, startVertex,out):
        startVertex.setColor('gray')
        self.time += 1
        startVertex.setDiscovery(self.time)
        for nextVertex in startVertex.getConnections():
            if nextVertex.getColor() == 'white':
                nextVertex.setPred(startVertex.getId())
                self.dfsvisit(nextVertex,out)
        startVertex.setColor('black')
        self.time += 1
        startVertex.setFinish(self.time)
        out.append(startVertex.getId())
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27

测试

在这里插入图片描述

if __name__ =='__main__':

    g = Graph()
    for i in range(1,9):
        g.addVertex(i)
    g.addEdge(1,2)
    g.addEdge(2,1)
    g.addEdge(1,5)
    g.addEdge(5,1)
    g.addEdge(2, 6)
    g.addEdge(6,2)
    g.addEdge(6, 3)
    g.addEdge(3,6)
    g.addEdge(6, 7)
    g.addEdge(7,6)
    g.addEdge(3, 7)
    g.addEdge(7,3)
    g.addEdge(3, 4)
    g.addEdge(4,3)
    g.addEdge(4, 8)
    g.addEdge(4,7)
    g.addEdge(7,4)
    g.addEdge(8,4)
    g.addEdge(7, 8)
    g.addEdge(8,7)


    g.bfs(g.vertList[2])
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()

    x = g.dfs()
    print(x)
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()
    for v in g:
        print(v.getDiscovery(),end = ' ')
    print()
    for v in g:
        print(v.getFinish(),end = ' ')
    print()
    #输出
    # 2 None 6 3 1 2 6 7 
    # 1 0 2 3 2 1 2 3

    # [8, 4, 7, 3, 6, 2, 5, 1]

    # None 1 6 7 1 2 3 4 
    # 1 0 2 3 2 1 2 3 
    # 1 2 4 6 14 3 5 7 
    # 16 13 11 9 15 12 10 8 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63

最小生成树

prim最小生成树

prim最小生成树伪代码

在这里插入图片描述

    def prim_simple(self,start = None):
        '''
        prim最小生成树简单版本
        :param start: 迭代起点
        :return:
        '''
        #在有向图中，有可能存在这样一种情况：两个节点之间来和回的权重不一样
        #所以prim在某些情况下会失效
        if start == None:
            # 若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)

        for i in range(self.numVertices):
            minDist = inf
            rec = -1
            for j in self:
                if j.getColor() == 'white' and j.getDistance() < minDist:
                    minDist = j.getDistance()
                    rec = j
                    
            for u in rec.getConnections():
                if rec.getWeight(u) < u.getDistance():
                    u.setDistance(rec.getWeight(u))
                    u.setPred(rec.getId())
                    
            rec.setColor('black')

            # for u in rec.getConnections():
            #     newCost = rec.getWeight(u) + rec.getDistance()
            #     if newCost < u.getDistance():
            #         u.setDistance(newCost)
            #         u.setPred(rec.getId())
            # rec.setColor('black')

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40

prim 最小生成树（优先对列）

在这里插入图片描述

    def prim(self,start = None):
        '''
        prim最小生成树（采用优先队列）
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)
        pq = queue.PriorityQueue()

        for v in list(self.getVertics()):

            pq.put((self.vertList[v].getDistance(),v))


        while not pq.empty():
            (t, currentVert) = pq.get()
            currentVert = self.vertList[currentVert]
            for nextVert in currentVert.getConnections():
                newCost = currentVert.getWeight(nextVert)
                if newCost < nextVert.getDistance() and nextVert.getColor() == 'white':
                    nextVert.setPred(currentVert.getId())
                    nextVert.setDistance(newCost)
                    pq.put((newCost, nextVert.id))
            currentVert.setColor('black')
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

kruskal最小生成树

在这里插入图片描述


class DisjointSet:

    def __init__(self, element=None):
        self._father = {}
        self._rank = {}
        # 初始化时每个元素单独成为一个集合
        if element is not None:
            for i in element:
                self.add(i)

    def add(self, x):
        # 添加新集合
        # 如果已经存在则跳过
        if x in self._father:
            return
        self._father[x] = x
        self._rank[x] = 0

    def _query(self, x):
        # 如果father[x] == x，说明x是树根
        if self._father[x] == x:
            return x
        self._father[x] = self._query(self._father[x])
        return self._father[x]

    def merge(self, x, y):
        if x not in self._father:
            self.add(x)
        if y not in self._father:
            self.add(y)
        # 查找到两个元素的树根
        x = self._query(x)
        y = self._query(y)
        # 如果相等，说明属于同一个集合
        if x == y:
            return
        # 否则将树深小的合并到树根大的上
        if self._rank[x] < self._rank[y]:
            self._father[x] = y
        else:
            self._father[y] = x
            # 如果树深相等，合并之后树深+1
            if self._rank[x] == self._rank[y]:
                self._rank[x] += 1

    # 判断是否属于同一个集合
    def same(self, x, y):
        return self._query(x) == self._query(y)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49

在这里插入图片描述

    def kruskal(self):
        '''
        kruskal算法使用并查集类
        :return:
        '''
        disjoinset = DisjointSet([key.getId() for key in self])
        edges = []
        for v in self:
            for u in v.getConnections():
                edges.append((v.getId(),u.getId(),v.getWeight(u)))

        edges.sort(key = lambda a:a[2])

        minu_tree = []
        for edge in edges:
            u, v, w = edge
            if disjoinset.same(u, v):
                continue
            disjoinset.merge(u, v)
            self.getVertex(v).setPred(u)
            minu_tree.append(edge)
        for edge in minu_tree:
            self.getVertex(u)

        return minu_tree
    
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

测试

在这里插入图片描述

if __name__ =='__main__':

    g = Graph()
    g.addVertex('a')
    g.addVertex('b')
    g.addVertex('c')
    g.addVertex('d')
    g.addVertex('f')
    g.addVertex('h')
    g.addVertex('g')
    g.addVertex('i')
    g.addVertex('z')
    g.addEdge('a','b',4)
    g.addEdge('b','a',4)
    g.addEdge('a','h',8)
    g.addEdge('h','a',8)
    g.addEdge('b','c',8)
    g.addEdge('c','b',8)
    g.addEdge('b','h',1)
    g.addEdge('h','b',1)
    g.addEdge('h','i',7)
    g.addEdge('i','h',7)
    g.addEdge('c','i',2)
    g.addEdge('i','c',2)
    g.addEdge('c','d',7)
    g.addEdge('d','c',7)
    g.addEdge('i','g',4)
    g.addEdge('g','i',4)
    g.addEdge('g','f',2)
    g.addEdge('f','g',2)
    g.addEdge('f','z',10)
    g.addEdge('z','f',10)
    g.addEdge('d','f',14)
    g.addEdge('f','d',14)
    g.addEdge('d','z',9)
    g.addEdge('z','d',9)
    g.addEdge('c','f',6)
    g.addEdge('f','c',6)
    g.addEdge('h','g',1)
    g.addEdge('g','h',1)
    g.prim()
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()

    g.prim_simple()
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()
    x = g.kruskal()
    print(x)
    #输出
    # None a i c g b h g d 
    # 0 4 2 7 2 1 1 4 9 

    # a,0
    # b,4
    # h,1
    # g,1
    # f,2
    # i,4
    # c,2
    # d,7
    # z,9

    # None h i c g b h c d 
    # 0 1 2 7 2 1 1 2 9

    #（这个输出在遍历时被覆盖掉了，只能通过print打印查找记录） 
    # [('b', 'h', 1), ('h', 'g', 1), ('c', 'i', 2), ('f', 'g', 2), ('a', 'b', 4), ('g', 'i', 4), ('c', 'd', 7), ('d', 'z', 9)]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80

单源最短路径

dijkstra单源最短路径

在这里插入图片描述

    def dijkstra_simple(self,start = None):
        '''
         dijkstra单源最短路径简单版本
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)

        for i in range(self.numVertices):
            minDist = inf
            rec = 0
            for j in self:
                if j.getColor() == 'white' and j.getDistance() < minDist:
                    minDist = j.getDistance()
                    rec = j
            for u in rec.getConnections():
                if rec.getDistance() + rec.getWeight(u) < u.getDistance():
                    u.setDistance(rec.getDistance() + rec.getWeight(u))
                    u.setPred(rec.getId())
            rec.setColor('black')

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29

dijkstra单源最短路径(优先队列)

在这里插入图片描述


    def dijkstra(self,start = None):
        '''
        dijkstra单源最短路径（使用优先队列）
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]


        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')

        start.setDistance(0)
        pq = queue.PriorityQueue()
        for v in list(self.getVertics()):
            pq.put((self.vertList[v].getDistance(),v))

        while not pq.empty():
            (t, currentVert) = pq.get()
            currentVert = self.vertList[currentVert]
            for nextVert in currentVert.getConnections():
                newDist = currentVert.getWeight(nextVert) + currentVert.getDistance()
                if newDist < nextVert.getDistance() and nextVert.getColor() == 'white':
                    nextVert.setPred(currentVert.getId())
                    nextVert.setDistance(newDist)
                    pq.put((newDist, nextVert.id))

            currentVert.setColor('black')

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34

测试

在这里插入图片描述

if __name__ =='__main__':

    g = Graph()
    g.addVertex('s')
    g.addVertex('t')
    g.addVertex('x')
    g.addVertex('y')
    g.addVertex('z')
    g.addEdge('s','t',8)
    g.addEdge('s','y',5)
    g.addEdge('t','x',1)
    g.addEdge('t','y',2)
    g.addEdge('x','z',4)
    g.addEdge('y','t',3)
    g.addEdge('y','z',2)
    g.addEdge('y','x',9)
    g.addEdge('z','x',6)
    g.dijkstra()
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()

    g.dijkstra_simple()
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

bellman_ford单源最短路径

在这里插入图片描述

   def bellman_ford(self,start = None):
        '''
        bellman_ford算法
        :param start:
        :return:
        '''
        if start == None:
            # 若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        #可以用这种方法遍历 不过比较繁琐。
        # for v in self.getVertics():
        #     self.vertList[v].setDistance(inf)
        #     self.vertList[v].setPred(None)
        #     pred[v] = None
        for v in self:
            v.setDistance(inf)
            v.setPred(None)


        start.setDistance(0)
        for i in range(self.numVertices-1):
            for v in self:
                for u in v.getConnections():
                    if v.getDistance() + v.getWeight(u) < u.getDistance():
                        u.setDistance(v.getDistance() + v.getWeight(u))
                        u.setPred(v.getId())


        for v in self:
            for u in v.getConnections():
                if v.getDistance() + v.getWeight(u) < u.getDistance():
                    print("存在负环")
                    return False
        return True
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35

测试

在这里插入图片描述


if __name__ =='__main__':

    bellman = Graph()
    bellman.addVertex('s')
    bellman.addVertex('t')
    bellman.addVertex('x')
    bellman.addVertex('y')
    bellman.addVertex('z')
    bellman.addEdge('s','t',6)
    bellman.addEdge('s','y',7)
    bellman.addEdge('t','x',5)
    bellman.addEdge('x','t',-2)
    bellman.addEdge('t','z',-4)
    bellman.addEdge('t','y',8)
    bellman.addEdge('y','z',2)
    bellman.addEdge('y','x',-3)
    bellman.addEdge('x','z',7)
    bellman.bellman_ford()
    for v in bellman:
        print(v)
    for v in bellman:
        print(v.getPred(),end = ' ')
    print()
    for v in bellman:
        print(v.getDistance(),end = ' ')
    print()
    #输出
    # None x y s t 
    # 0 2 4 7 -2 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

图的遍历的应用

有向图的环路存在性判断

在这里插入图片描述


    def dfs_judge_cycle(self):

        for aVertex in self:
            aVertex.setColor('white')
            aVertex.setPred(-1)
        for aVertex in self:
            if aVertex.getColor() == 'white':
                if self.dfs_visit_judge_cycle(aVertex) == True:
                    return True
        return False

    def dfs_visit_judge_cycle(self, startVertex):
        startVertex.setColor('gray')
        startVertex.setDiscovery(self.time)
        for nextVertex in startVertex.getConnections():
            if nextVertex.getColor() == 'gray':
                return True
            if nextVertex.getColor() == 'white':
                nextVertex.setPred(startVertex.getId())
                if self.dfs_visit_judge_cycle(nextVertex) == True:
                    return True

            startVertex.setColor('black')
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

拓扑排序

广度优先策略

在这里插入图片描述

    #入度出度计算方法
    def getOutDegrees(self,vertex):
        '''
        获取某个顶点出度
        :param vertex:
        :return:
        '''

        return self.vertList[vertex].OutDegree

    def getInDegrees(self,vertex):
        '''
        获取某个顶点的入度（很蠢的一个做法哈哈）
        :param vertex:
        :return:
        '''
        #初始化完后遍历查找
        self.vertList[vertex].InDegree = 0
        for u in self:
            if self.vertList[vertex] in u.getConnections():
                self.vertList[vertex].InDegree += 1

        return self.vertList[vertex].InDegree
    def OutDegreeUpdate(self):
        for u in self:
            self.OutDegreeList[u.getId()] = len(u.getConnections())
    def InDegreeUpdate(self):
        for u in self:
            self.InDegreeList[u.getId()] = 0

        for u in self:
            for v in u.getConnections():
                self.InDegreeList[v.getId()] += 1

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34

    def topological_sort_bfs(self):

        self.InDegreeUpdate()
        vertQueue = queue.Queue()
        out = []

        for v in self:
            if self.InDegreeList[v.getId()] == 0:
                vertQueue.put(v)

        while vertQueue.empty() == False:
            currentVert = vertQueue.get()
            # print(currentVert.getId(),end=" ")
            out.append(currentVert.getId())
            for nbr in currentVert.getConnections():
                self.InDegreeList[nbr.getId()] = self.InDegreeList[nbr.getId()] - 1
                if self.InDegreeList[nbr.getId()] == 0:
                    vertQueue.put(nbr)
        # print()
        # print(out)
        return out


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

深度优先策略

在这里插入图片描述

    def topological_sort_dfs(self):

        self.out = []
        L = []
        for aVertex in self:
            aVertex.setColor('white')
        for aVertex in self:
            if aVertex.getColor() == 'white':
                self.topological_sort_dfs_visit(aVertex, L)
        L.reverse()
        return L

    def topological_sort_dfs_visit(self, startVertex, L):

        startVertex.setColor('gray')

        for nextVertex in startVertex.getConnections():
            if nextVertex.getColor() == 'white':
                self.topological_sort_dfs_visit(nextVertex, L)
        startVertex.setColor('black')
        L.append(startVertex.getId())
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

测试

在这里插入图片描述

if __name__ == '__main__':

    #拓扑排序测试
    topological = Graph()
    topological.addVertex('短裤')
    topological.addVertex('长裤')
    topological.addVertex('腰带')
    topological.addVertex('衬衫')
    topological.addVertex('领带')
    topological.addVertex('外套')
    topological.addVertex('袜子')
    topological.addVertex('鞋')
    topological.addVertex('手表')
    topological.addEdge('袜子','鞋')
    topological.addEdge('短裤','鞋')
    topological.addEdge('短裤','长裤')
    topological.addEdge('长裤','鞋')
    topological.addEdge('长裤','腰带')
    topological.addEdge('衬衫','腰带')
    topological.addEdge('衬衫','领带')
    topological.addEdge('领带','外套')
    topological.addEdge('腰带','外套')
    x = topological.topological_sort_bfs()
    print(x)
    x = topological.topological_sort_dfs()
    print(x)
    #输出
    #['短裤', '衬衫', '袜子', '手表', '长裤', '领带', '鞋', '腰带', '外套']
    #['手表', '袜子', '衬衫', '领带', '短裤', '长裤', '腰带', '外套', '鞋']
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29

强连通分量

    #逆序图
    def reverse(self):
        new = Graph()
        for u in self:
            new.addVertex(u.getId())
        for u in self:
            for v in u.getConnections():
                new.addEdge(v.getId(),u.getId(),u.getWeight(v))
        new.InDegreeUpdate()
        new.OutDegreeUpdate()
        return new
1
2
3
4
5
6
7
8
9
10
11

    def Strongly_Connected_Component(self):
        R = []
        G_r = self.reverse()
        L = G_r.dfs()
        for i in G_r:
            i.setColor('white')
        L_scc = []
        L.reverse()
        for i in L:
            if self.vertList[i].getColor() == 'white':
                self.dfsvisit(self.getVertex(i), L_scc)
                R.append(L_scc)
                L_scc = []
        return R
1
2
3
4
5
6
7
8
9
10
11
12
13
14

测试

在这里插入图片描述

if __name__ == '__main__':
    #强联通分量测试
    scc = Graph()
    for i in range(1,11):
        scc.addVertex(i)
    scc.addEdge(1,10)
    scc.addEdge(10,1)
    scc.addEdge(1,3)
    scc.addEdge(3,4)
    scc.addEdge(4,1)
    scc.addEdge(8,10)
    scc.addEdge(8,9)
    scc.addEdge(9,8)
    scc.addEdge(9,7)
    scc.addEdge(7,7)
    scc.addEdge(3,7)
    scc.addEdge(4,6)
    scc.addEdge(6,5)
    scc.addEdge(5,2)
    scc.addEdge(2,6)
    R = scc.Strongly_Connected_Component()
    print(R)
    #输出
    #[[7], [5, 6, 2], [10, 4, 3, 1], [9, 8]]

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

后记

主要是巩固一下python的编程能力，顺便复习一下数据结构与算法。
后续有精力把算法导论里的算法补全。
创造不易，请多多支持。

参考书目

1.中国大学MOOC北航《算法设计与分析》
2.Python数据结构与算法分析（第二版）

整体代码

import queue
import sys
import os
import unittest

inf = 65535
GRAY = 0
WHITE = -1
BLACK = 1


class DisjointSet:

    def __init__(self, element=None):
        self._father = {}
        self._rank = {}
        # 初始化时每个元素单独成为一个集合
        if element is not None:
            for i in element:
                self.add(i)

    def add(self, x):
        # 添加新集合
        # 如果已经存在则跳过
        if x in self._father:
            return
        self._father[x] = x
        self._rank[x] = 0

    def _query(self, x):
        # 如果father[x] == x，说明x是树根
        if self._father[x] == x:
            return x
        self._father[x] = self._query(self._father[x])
        return self._father[x]

    def merge(self, x, y):
        if x not in self._father:
            self.add(x)
        if y not in self._father:
            self.add(y)
        # 查找到两个元素的树根
        x = self._query(x)
        y = self._query(y)
        # 如果相等，说明属于同一个集合
        if x == y:
            return
        # 否则将树深小的合并到树根大的上
        if self._rank[x] < self._rank[y]:
            self._father[x] = y
        else:
            self._father[y] = x
            # 如果树深相等，合并之后树深+1
            if self._rank[x] == self._rank[y]:
                self._rank[x] += 1

    # 判断是否属于同一个集合
    def same(self, x, y):
        return self._query(x) == self._query(y)



class Vertex:
    '''
    创建一个空顶点
    '''
    def __init__(self,key):
        self.id = key
        self.connectedTo = {}
        self.color = 'white'
        self.dist = inf
        self.pred = None
        self.InDegree = 0
        self.OutDegree = 0

        self.disc = 0
        self.fin = 0
    def addNeighbor(self, nbr, weight = 0):
        '''
        添加邻接的顶点
        :param nbr: 邻接的顶点
        :param weight: 默认为0
        :return:
        '''
        self.connectedTo[nbr] = weight
        self.OutDegree += 1

    def getConnections(self):
        '''
        获取邻接的所以顶点
        :return:
        '''
        return self.connectedTo.keys()
    def getId(self):
        '''
        获取自身顶点的值
        :return:
        '''
        return self.id
    def getWeight(self, nbr):
        '''
        两顶点连接边的权重
        :param nbr:
        :return:
        '''
        return self.connectedTo[nbr]

    def setColor(self, color):
        '''
        设置顶点的颜色（一般设置为白色表示未访问，灰色表示待处理，黑色表示访问完成）。
        :param color: 字符串、或者int 例如 color = 'white' color = -1
        :return:
        '''
        self.color = color

    def getColor(self):
        '''
        获取顶点的颜色，（一般设置为白色表示未访问，灰色表示待处理，黑色表示访问完成）
        :return:
        '''
        return self.color

    def setDistance(self, d):
        '''
        设置顶点距离，（不同算法有不同的意义）如单源最短路径表示源点到该点最短距离，最小生成树表示离该顶点最小的权重。
        :param d:
        :return:
        '''
        self.dist = d

    def getDistance(self):
        '''
        获取该顶点的距离，（不同算法有不同的意义）如单源最短路径表示源点到该点最短距离，最小生成树表示离该顶点最小的权重。
        :return:
        '''
        return self.dist


    def setPred(self, p):
        '''
        设置该顶点的前驱顶点
        :param p:
        :return:
        '''
        self.pred = p

    def getPred(self):
        '''
        设置该顶点的前驱顶点
        :return:
        '''
        return self.pred


    ##以下方法用得不多，帮助理解。

    def setDiscovery(self, dtime):
        '''
        设置顶点发现时间。
        :param dtime:
        :return:
        '''
        self.disc = dtime
    def getDiscovery(self):
        '''
        获取顶点发现时间。
        :return:
        '''
        return self.disc

    def setFinish(self, ftime):
        '''
        设置顶点完成时间。
        :param ftime:
        :return:
        '''
        self.fin = ftime

    def getFinish(self):
        '''
        获取顶点完成时间。
        :return:
        '''
        return self.fin









    # def __str__(self):
    #     return str(self.id) + ":color " + self.color + ":disc " + str(self.disc) + ":fin " + str(
    #         self.fin) + ":dist " + str(self.dist) + ":pred \n\t[" + str(self.pred) + "]\n"
    def __str__(self):
        return str(self.id) + ":color " + self.color + ":disc " + str(self.disc) + ":fin " + str(
            self.fin) + ":dist " + str(self.dist) + ":pred \n\t[" + str(self.pred) + "]\n"




class Graph:
    '''
    新建一个空图。
    '''
    def __init__(self):
        '''
        初始化邻接表，顶点个数
        '''
        self.vertList = {}
        self.numVertices = 0
        self.time = 0
        self.InDegreeList = {}
        self.OutDegreeList = {}
    def addVertex(self, key):
        '''
        向图中添加一个顶点实例
        :param key:
        :return: None
        '''
        self.numVertices = self.numVertices + 1
        newVertex = Vertex(key)
        self.vertList[key] = newVertex
        return newVertex

    def getVertex(self, n):
        '''
        在图中找到名为n的顶点
        :param n:
        :return:
        '''
        if n in self.vertList:
            return self.vertList[n]
        else:
            return None
    def __contains__(self,n):
        '''
        对in进行重载
        :param n:
        :return:
        '''
        return n in self.vertList
    def addEdge(self, f, t, weight = 0):
        '''
        )向图中添加一条带权重cost的有向边，用于连接顶点f和t
        :param f:
        :param t:
        :param weight:
        :return: None
        '''
        if f not in self.vertList:
            nv = self.addVertex(f)
        if t not in self.vertList:
            nv = self.addVertex(t)
        self.vertList[f].addNeighbor(self.vertList[t],weight)

    def getVertics(self):
        '''
        以列表形式返回图中所有顶点
        :return: {}.keys()
        '''
        return self.vertList.keys()
    def __iter__(self):
        '''
        对迭代进行重载，方便遍历顶点邻接表
        :return:
        '''
        return iter(self.vertList.values())

    def getOutDegrees(self,vertex):
        '''
        获取某个顶点出度
        :param vertex:
        :return:
        '''

        return self.vertList[vertex].OutDegree

    def getInDegrees(self,vertex):
        '''
        获取某个顶点的入度（很蠢的一个做法哈哈）
        :param vertex:
        :return:
        '''
        #初始化完后遍历查找
        self.vertList[vertex].InDegree = 0
        for u in self:
            if self.vertList[vertex] in u.getConnections():
                self.vertList[vertex].InDegree += 1

        return self.vertList[vertex].InDegree
    def OutDegreeUpdate(self):
        for u in self:
            self.OutDegreeList[u.getId()] = len(u.getConnections())
    def InDegreeUpdate(self):
        for u in self:
            self.InDegreeList[u.getId()] = 0

        for u in self:
            for v in u.getConnections():
                self.InDegreeList[v.getId()] += 1




    def bfs(self,start = None):
        '''
        广度优先搜索
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]
        start.setDistance(0)
        start.setPred(None)
        vertQueue = queue.Queue()
        vertQueue.put(start)
        out = []
        while vertQueue.empty() == False:
            currentVert = vertQueue.get()
            for nbr in currentVert.getConnections():
                if nbr.getColor() == 'white':
                    nbr.setColor('gray')
                    nbr.setDistance(currentVert.getDistance() + 1)
                    nbr.setPred(currentVert.getId())
                    vertQueue.put(nbr)
            currentVert.setColor('black')
            out.append(currentVert.getId())
        return out

    def dfs(self):
        '''
        深度优先搜索
        :return:
        '''
        out = []
        self.time = 0
        for aVertex in self:
            aVertex.setColor('white')
            aVertex.setPred(None)
        for aVertex in self:
            if aVertex.getColor() == 'white':
                self.dfsvisit(aVertex,out)
        return out

    def dfsvisit(self, startVertex,out):
        startVertex.setColor('gray')
        self.time += 1
        startVertex.setDiscovery(self.time)
        for nextVertex in startVertex.getConnections():
            if nextVertex.getColor() == 'white':
                nextVertex.setPred(startVertex.getId())
                self.dfsvisit(nextVertex,out)
        startVertex.setColor('black')
        self.time += 1
        startVertex.setFinish(self.time)
        out.append(startVertex.getId())


    def dfs_stack(self,start = None):
        '''
        用栈实现dfs 该方法在某些情况下会失效 比如遍历生成拓扑排序
        :param start:
        :return:
        '''
        from collections import deque
        stack = deque()
        if start == None:
            # 若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setColor('white')
            v.setPred(None)
        stack.append(start)

        while len(stack) > 0:
            currentVert = stack.pop()
            for nbr in currentVert.getConnections():
                if nbr.getColor() == 'white':
                    stack.append(nbr)
                    nbr.setPred(currentVert.getId())
                    nbr.setColor('balck')


    def prim_simple(self,start = None):
        '''
        prim最小生成树简单版本
        :param start: 迭代起点
        :return:
        '''
        #在有向图中，有可能存在这样一种情况：两个节点之间来和回的权重不一样
        #所以prim在某些情况下会失效
        if start == None:
            # 若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)

        for i in range(self.numVertices):
            minDist = inf
            rec = -1
            # rec = 0
            for j in self:
                if j.getColor() == 'white' and j.getDistance() < minDist:
                    minDist = j.getDistance()
                    rec = j
            print(('%s,%d')%(rec.getId(),rec.getDistance()))
            for u in rec.getConnections():
                if rec.getWeight(u) < u.getDistance():
                    u.setDistance(rec.getWeight(u))
                    # print(('%s,%s,%d')%(rec.getId(),u.getId(),rec.getWeight(u)))
                    u.setPred(rec.getId())

            rec.setColor('black')

            # for u in rec.getConnections():
            #     newCost = rec.getWeight(u) + rec.getDistance()
            #     if newCost < u.getDistance():
            #         u.setDistance(newCost)
            #         u.setPred(rec.getId())
            # rec.setColor('black')


    def prim(self,start = None):
        '''
        prim最小生成树（采用优先队列）
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)
        pq = queue.PriorityQueue()

        for v in list(self.getVertics()):

            pq.put((self.vertList[v].getDistance(),v))


        while not pq.empty():
            (t, currentVert) = pq.get()
            currentVert = self.vertList[currentVert]
            for nextVert in currentVert.getConnections():
                newCost = currentVert.getWeight(nextVert)
                if newCost < nextVert.getDistance() and nextVert.getColor() == 'white':
                    nextVert.setPred(currentVert.getId())
                    nextVert.setDistance(newCost)
                    pq.put((newCost, nextVert.id))
            currentVert.setColor('black')


    def dijkstra(self,start = None):
        '''
        dijkstra单源最短路径（使用优先队列）
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]


        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')

        start.setDistance(0)
        pq = queue.PriorityQueue()
        for v in list(self.getVertics()):
            pq.put((self.vertList[v].getDistance(),v))

        while not pq.empty():
            (t, currentVert) = pq.get()
            currentVert = self.vertList[currentVert]
            for nextVert in currentVert.getConnections():
                newDist = currentVert.getWeight(nextVert) + currentVert.getDistance()
                if newDist < nextVert.getDistance() and nextVert.getColor() == 'white':
                    nextVert.setPred(currentVert.getId())
                    nextVert.setDistance(newDist)
                    pq.put((newDist, nextVert.id))

            currentVert.setColor('black')
    def dijkstra_simple(self,start = None):
        '''
         dijkstra单源最短路径简单版本
        :param start:
        :return:
        '''
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)

        for i in range(self.numVertices):
            minDist = inf
            rec = 0
            for j in self:
                if j.getColor() == 'white' and j.getDistance() < minDist:
                    minDist = j.getDistance()
                    rec = j
            for u in rec.getConnections():
                if rec.getDistance() + rec.getWeight(u) < u.getDistance():
                    u.setDistance(rec.getDistance() + rec.getWeight(u))
                    u.setPred(rec.getId())
            rec.setColor('black')


    def bellman_ford(self,start = None):
        '''
        bellman_ford算法
        :param start:
        :return:
        '''
        if start == None:
            # 若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        #可以用这种方法遍历 不过比较繁琐。
        # for v in self.getVertics():
        #     self.vertList[v].setDistance(inf)
        #     self.vertList[v].setPred(None)
        #     pred[v] = None
        for v in self:
            v.setDistance(inf)
            v.setPred(None)


        start.setDistance(0)
        for i in range(self.numVertices-1):
            for v in self:
                for u in v.getConnections():
                    print("(%s,%s,%s)" % (v.getId(), u.getId(), v.getWeight(u)))
                    if v.getDistance() + v.getWeight(u) < u.getDistance():
                        u.setDistance(v.getDistance() + v.getWeight(u))
                        u.setPred(v.getId())


        for v in self:
            for u in v.getConnections():
                if v.getDistance() + v.getWeight(u) < u.getDistance():
                    print("存在负环")
                    return False
                    break
        return True


    def kruskal(self):
        '''
        kruskal算法使用并查集类
        :return:
        '''
        disjoinset = DisjointSet([key.getId() for key in self])
        edges = []
        for v in self:
            for u in v.getConnections():
                edges.append((v.getId(),u.getId(),v.getWeight(u)))

        edges.sort(key = lambda a:a[2])

        minu_tree = []
        for edge in edges:
            u, v, w = edge
            if disjoinset.same(u, v):
                continue
            disjoinset.merge(u, v)
            self.getVertex(v).setPred(u)
            minu_tree.append(edge)
        for edge in minu_tree:
            self.getVertex(u)

        return minu_tree

    def dfs_judge_cycle(self):

        for aVertex in self:
            aVertex.setColor('white')
            aVertex.setPred(-1)
        for aVertex in self:
            if aVertex.getColor() == 'white':
                if self.dfs_visit_judge_cycle(aVertex) == True:
                    return True
        return False

    def dfs_visit_judge_cycle(self, startVertex):
        startVertex.setColor('gray')
        startVertex.setDiscovery(self.time)
        for nextVertex in startVertex.getConnections():
            if nextVertex.getColor() == 'gray':
                return True
            if nextVertex.getColor() == 'white':
                nextVertex.setPred(startVertex.getId())
                if self.dfs_visit_judge_cycle(nextVertex) == True:
                    return True

            startVertex.setColor('black')

    def topological_sort_bfs(self):

        self.InDegreeUpdate()
        vertQueue = queue.Queue()
        out = []

        for v in self:
            if self.InDegreeList[v.getId()] == 0:
                vertQueue.put(v)

        while vertQueue.empty() == False:
            currentVert = vertQueue.get()
            # print(currentVert.getId(),end=" ")
            out.append(currentVert.getId())
            for nbr in currentVert.getConnections():
                self.InDegreeList[nbr.getId()] = self.InDegreeList[nbr.getId()] - 1
                if self.InDegreeList[nbr.getId()] == 0:
                    vertQueue.put(nbr)
        # print()
        # print(out)
        return out

    def topological_sort_dfs(self):

        self.out = []
        L = []
        for aVertex in self:
            aVertex.setColor('white')
        for aVertex in self:
            if aVertex.getColor() == 'white':
                self.topological_sort_dfs_visit(aVertex, L)
        L.reverse()
        return L

    def topological_sort_dfs_visit(self, startVertex, L):

        startVertex.setColor('gray')

        for nextVertex in startVertex.getConnections():
            if nextVertex.getColor() == 'white':
                self.topological_sort_dfs_visit(nextVertex, L)
        startVertex.setColor('black')
        L.append(startVertex.getId())

    def Strongly_Connected_Component(self):
        R = []
        G_r = self.reverse()
        L = G_r.dfs()
        for i in G_r:
            i.setColor('white')
        L_scc = []
        L.reverse()
        for i in L:
            if self.vertList[i].getColor() == 'white':
                self.dfsvisit(self.getVertex(i), L_scc)
                R.append(L_scc)
                L_scc = []
        return R

    def reverse(self):
        new = Graph()
        for u in self:
            new.addVertex(u.getId())
        for u in self:
            for v in u.getConnections():
                new.addEdge(v.getId(),u.getId(),u.getWeight(v))
        new.InDegreeUpdate()
        new.OutDegreeUpdate()
        return new

    ###以下内容学习副产物 可以略过
    def prim_other(self,start = None):
        if start == None:
            #若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        for v in self:
            v.setDistance(inf)
            v.setPred(None)
            v.setColor('white')
        start.setDistance(0)
        pq = queue.PriorityQueue()

        for v in list(self.getVertics()):

            pq.put((self.vertList[v].getDistance(),v))


        while not pq.empty():
            (t, currentVert) = pq.get()
            currentVert = self.vertList[currentVert]
            for nextVert in currentVert.getConnections():
                newCost = currentVert.getWeight(nextVert) + currentVert.getDistance()
                if newCost < nextVert.getDistance() and nextVert.getColor() == 'white':
                    nextVert.setPred(currentVert.getId())
                    nextVert.setDistance(newCost)
                    pq.put((newCost, nextVert.id))
            currentVert.setColor('black')



    def BFS(self, s = None):
        if s == None:
            #若是未指定远点 默认取第一个顶点
            s = list(self.getVertics())[0]
        color = {}
        pred = {}
        dist = {}
        queue = []
        out = []
        for i in self.vertList:
            color[i] = WHITE
            dist[i] = inf
            pred[i] = -1

        color[s] = GRAY
        dist[s] = 0
        queue.append(s)
        while queue:
            u = queue.pop(0)
            for i in list(self.vertList[u].getConnections()):
                print(i)
                if color[i.id] == WHITE:
                    color[i.id] = GRAY
                    dist[i.id] = dist[u] + 1
                    pred[i.id] = u
                    queue.append(i.id)

            color[u] = BLACK
            out.append(u)

        print(out)
        print(pred.values())
        print(dist.values())
        return out,pred,dist

    def kruskal_simple(self):
        '''
        kruskal算法简约写法（未用类）
        :return:
        '''
        X = dict()
        R = dict()  # 各点的初始等级均为0,如果被做为连接的的末端，则增加1

        def make_set(point):
            X[point] = point
            R[point] = 0

        def find(point):
            if X[point] != point:
                X[point] = find(X[point])
            return X[point]

        def merge(point1, point2):
            '''连接两个分量（节点）
            '''
            r1 = find(point1)
            r2 = find(point2)
            if r1 != r2:
                if R[r1] > R[r2]:
                    X[r2] = r1
                else:
                    X[r1] = r2
                    if R[r1] == R[r2]:
                        R[r2] += 1

        edges = []
        for v in self:
            for u in v.getConnections():
                edges.append((v.getId(), u.getId(), v.getWeight(u)))
        print(edges)

        edges.sort(key=lambda a: a[2])
        print(edges)  # 按照权重从小到大排序

        for v in g:
            make_set(v.getId())
        minu_tree = []
        for edge in edges:
            v, u, weight = edge
            if find(v) != find(u):
                merge(v, u)
                self.getVertex(u).setPred(v)
                minu_tree.append(edge)
        return minu_tree


    def topological_sort_dfs_stack(self,start = None):
        from collections import deque
        stack = deque()
        if start == None:
            # 若是未指定远点 默认取第一个顶点
            start = self.vertList[list(self.getVertics())[0]]

        L = []
        for v in self:
            v.setColor('white')
        stack.append(start)

        while len(stack) > 0:
            currentVert = stack.pop()
            print(currentVert.getId())
            for nbr in currentVert.getConnections():
                print(nbr.getId())
                if nbr.getColor() == 'white':
                    stack.append(nbr)
                    nbr.setColor('balck')
            L.append(currentVert.getId())
        return L




def traverse(y):
    x = y
    while x.getPred():
        print(x.getId())
        x = x.getPred()
    print(x.getId)




# if __name__ == '__main__':
#     #强联通分量测试
#     scc = Graph()
#     for i in range(1,11):
#         scc.addVertex(i)
#     scc.addEdge(1,10)
#     scc.addEdge(10,1)
#     scc.addEdge(1,3)
#     scc.addEdge(3,4)
#     scc.addEdge(4,1)
#     scc.addEdge(8,10)
#     scc.addEdge(8,9)
#     scc.addEdge(9,8)
#     scc.addEdge(9,7)
#     scc.addEdge(7,7)
#     scc.addEdge(3,7)
#     scc.addEdge(4,6)
#     scc.addEdge(6,5)
#     scc.addEdge(5,2)
#     scc.addEdge(2,6)
#     R = scc.Strongly_Connected_Component()
#     print(R)

# if __name__ == '__main__':
#
#     #拓扑排序测试
#     topological = Graph()
#     topological.addVertex('短裤')
#     topological.addVertex('长裤')
#     topological.addVertex('腰带')
#     topological.addVertex('衬衫')
#     topological.addVertex('领带')
#     topological.addVertex('外套')
#     topological.addVertex('袜子')
#     topological.addVertex('鞋')
#     topological.addVertex('手表')
#     topological.addEdge('袜子','鞋')
#     topological.addEdge('短裤','鞋')
#     topological.addEdge('短裤','长裤')
#     topological.addEdge('长裤','鞋')
#     topological.addEdge('长裤','腰带')
#     topological.addEdge('衬衫','腰带')
#     topological.addEdge('衬衫','领带')
#     topological.addEdge('领带','外套')
#     topological.addEdge('腰带','外套')
#     x = topological.topological_sort_bfs()
#     print(x)
#     x = topological.topological_sort_dfs()
#     print(x)

# if __name__ =='__main__':
#
#     bellman = Graph()
#     bellman.addVertex('s')
#     bellman.addVertex('t')
#     bellman.addVertex('x')
#     bellman.addVertex('y')
#     bellman.addVertex('z')
#     bellman.addEdge('s','t',6)
#     bellman.addEdge('s','y',7)
#     bellman.addEdge('t','x',5)
#     bellman.addEdge('x','t',-2)
#     bellman.addEdge('t','z',-4)
#     bellman.addEdge('t','y',8)
#     bellman.addEdge('y','z',2)
#     bellman.addEdge('y','x',-3)
#     bellman.addEdge('x','z',7)
#     bellman.bellman_ford()
#     for v in bellman:
#         print(v)
#     for v in bellman:
#         print(v.getPred(),end = ' ')
#     print()
#     for v in bellman:
#         print(v.getDistance(),end = ' ')
#     print()

# if __name__ =='__main__':
#
#     g = Graph()
#     g.addVertex('s')
#     g.addVertex('t')
#     g.addVertex('x')
#     g.addVertex('y')
#     g.addVertex('z')
#     g.addEdge('s','t',8)
#     g.addEdge('s','y',5)
#     g.addEdge('t','x',1)
#     g.addEdge('t','y',2)
#     g.addEdge('x','z',4)
#     g.addEdge('y','t',3)
#     g.addEdge('y','z',2)
#     g.addEdge('y','x',9)
#     g.addEdge('z','x',6)
#     g.dijkstra()
#     for v in g:
#         print(v)
#     for v in g:
#         print(v.getPred(),end = ' ')
#     print()
#     for v in g:
#         print(v.getDistance(),end = ' ')
#     print()
#
#     g.dijkstra_simple()
#     for v in g:
#         print(v)
#     for v in g:
#         print(v.getPred(),end = ' ')
#     print()
#     for v in g:
#         print(v.getDistance(),end = ' ')
#     print()

# if __name__ =='__main__':
#
#     g = Graph()
#     g.addVertex('a')
#     g.addVertex('b')
#     g.addVertex('c')
#     g.addVertex('d')
#     g.addVertex('f')
#     g.addVertex('h')
#     g.addVertex('g')
#     g.addVertex('i')
#     g.addVertex('z')
#     g.addEdge('a','b',4)
#     g.addEdge('b','a',4)
#     g.addEdge('a','h',8)
#     g.addEdge('h','a',8)
#     g.addEdge('b','c',8)
#     g.addEdge('c','b',8)
#     g.addEdge('b','h',1)
#     g.addEdge('h','b',1)
#     g.addEdge('h','i',7)
#     g.addEdge('i','h',7)
#     g.addEdge('c','i',2)
#     g.addEdge('i','c',2)
#     g.addEdge('c','d',7)
#     g.addEdge('d','c',7)
#     g.addEdge('i','g',4)
#     g.addEdge('g','i',4)
#     g.addEdge('g','f',2)
#     g.addEdge('f','g',2)
#     g.addEdge('f','z',10)
#     g.addEdge('z','f',10)
#     g.addEdge('d','f',14)
#     g.addEdge('f','d',14)
#     g.addEdge('d','z',9)
#     g.addEdge('z','d',9)
#     g.addEdge('c','f',6)
#     g.addEdge('f','c',6)
#     g.addEdge('h','g',1)
#     g.addEdge('g','h',1)
#     g.prim()
#     for v in g:
#         print(v)
#     for v in g:
#         print(v.getPred(),end = ' ')
#     print()
#     for v in g:
#         print(v.getDistance(),end = ' ')
#     print()
#
#     g.prim_simple()
#     for v in g:
#         print(v)
#     for v in g:
#         print(v.getPred(),end = ' ')
#     print()
#     for v in g:
#         print(v.getDistance(),end = ' ')
#     print()
#     x = g.kruskal()
#     print(x)

if __name__ =='__main__':

    g = Graph()
    for i in range(1,9):
        g.addVertex(i)
    g.addEdge(1,2)
    g.addEdge(2,1)
    g.addEdge(1,5)
    g.addEdge(5,1)
    g.addEdge(2, 6)
    g.addEdge(6,2)
    g.addEdge(6, 3)
    g.addEdge(3,6)
    g.addEdge(6, 7)
    g.addEdge(7,6)
    g.addEdge(3, 7)
    g.addEdge(7,3)
    g.addEdge(3, 4)
    g.addEdge(4,3)
    g.addEdge(4, 8)
    g.addEdge(4,7)
    g.addEdge(7,4)
    g.addEdge(8,4)
    g.addEdge(7, 8)
    g.addEdge(8,7)


    g.bfs(g.vertList[2])
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()

    x = g.dfs()
    print(x)
    for v in g:
        print(v)
    for v in g:
        print(v.getPred(),end = ' ')
    print()
    for v in g:
        print(v.getDistance(),end = ' ')
    print()
    for v in g:
        print(v.getDiscovery(),end = ' ')
    print()
    for v in g:
        print(v.getFinish(),end = ' ')
    print()
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068

相关阅读:
Python遥感开发之批量掩膜和裁剪
 华为u2000v200r018 同步网元失败提示服务器不可达解决办法
 基于混合整数遗传算法的最优成分选择（Matlab代码实现）
Django模板层
 python+allure+jenkins
golang中的select原理解析
 操作系统内存换入-请求调页---14
MonkeyRunner操作
 vue相关面试题:diff算法
 HK32F030MF4P6 SWD管脚功能复用GPIO
原文地址：https://blog.csdn.net/qq_30759585/article/details/126454566

全网最全Python图算法

文章目录